Complex Solutions for the Fisher Equation and the Benjamin-Bona-Mahony Equation

Transcription

1 Çankaya Unversty Journal of Scence and Engneerng Volume 7 (2010) No Complex Solutons for the Fsher Equaton and the Benjamn-Bona-Mahony Equaton Bülent Kılıç 1 and Erdal Baş 1 1 Department of Mathematcs Fırat Unversty Elazığ Turkey Correspondng author: bulent klc 27@hotmal.com Özet. Bu çalışmada Fsher ve Benjamn-Bona-Mahony denklemlernn karmaşık çözümler çn drekt cebrsel metodu sunulur. Bu metot kullanılarak Fsher ve Benjamn-Bona- Mahony denklemlernn bazı karmaşık çözümler elde edlr. Anahtar Kelmeler. Fsher denklem Benjamn-Bona-Mahony denklem drekt cebrsel metot karmaşık çözümler hareket eden dalga çözümler. Abstract. In ths artcle we gve drect algebrac method for the complex solutons of the Fsher equaton and Benjamn-Bona-Mahony equaton. We get some complex solutons of the Fsher equaton and Benjamn-Bona-Mahony equaton by ths method. Keywords. Fsher equaton Benjamn-Bona-Mahony equaton drect algebrac method complex solutons travelng wave solutons. 1. Introducton The theory of nonlnear dspersve wave moton s an nterestng area nvestgated n numerous artcles n whch t appears n relaton to varous subjects. We do not attempt to characterze the general form of nonlnear dspersve wave equatons [1 2]. These studes for nonlnear partal dfferental equatons have attracted much attenton n mathematcal physcs and play a crucal role n appled mathematcs. Furthermore when an orgnal nonlnear equaton s drectly calculated the soluton wll be n accord wth the physcal characterstcs of the actual phenomena [3]. Explct solutons to the nonlnear equatons are of fundamental mportance. Also dfferent methods for acqurng explct solutons to nonlnear evoluton equatons have been suggested. Many analytcal and numercal methods have been establshed n [4-27]. We may lst such examples as the generalzed Mura transformaton Darboux transformaton Cole-Hopf transformaton Hrota s dependent varable transformaton the nverse scatterng transform and the Bäcklund transformaton the Receved May ; accepted September ISSN c 2010 Çankaya Unversty

2 88 Kılıç and Baş tanh method sne-cosne method Panlevé method homogeneous balance method smlarty reducton method mproved tanh method etc. In [12] Parkes and Duffy have recently constructed an automated tanh-functon method. The authors present a Mathematca package that s concerned wth complcated algebra and outputs drectly the requred solutons for partcular nonlnear equatons. In ths paper our am s to fnd exact solutons of nonlnear PDE s especally complex solutons. In ths artcle the frst secton presents the scope of the study as an ntroducton. The second secton contans an analyss of the method gven n [24]. In the thrd secton we apply the method gven n [24] to the nonlnear Fsher equaton and the Benjamn-Bona-Mahony equaton. In the last secton we present the concluson. 2. An Analyss of the Method and Applcatons Frstly we wll gve a smple descrpton of the drect algebrac method [24]. For ths one can consder the general form of the nonlnear PDE n two varables Q(u u t u x u xx...) = 0 (1) and transform (1) wth u(x t) = u(ξ) ξ = k(x ct) where k and c are real constants. After transformaton we get a nonlnear ODE for u(ξ) Q (u kcu ku k 2 u...) = 0 (2) where u = du dξ. The soluton of (2) we are lookng for s expressed n the form n u(ξ) = a m F m (ξ) (3) m=0 where ξ = k(x ct) (where k and c are real constants) n s a postve nteger that can be determned by balancng the hghest order dervatve wth the hghest nonlnear terms n the equaton and a m and ξ can be determned. Substtutng (3) nto (2) yelds a set of algebrac equatons for F m (m = ) then all coeffcents of F m wll vansh. After ths separated algebrac equaton we fnd the coeffcents a 0 a m and ξ. F (ξ) expresses the soluton of the auxlary ordnary dfferental equaton where F = df dξ F (ξ) = + F 2 (ξ) and s a constant. Some solutons were gven n [24].

3 CUJSE 7 (2010) No In ths work we wll consder complex solutons of the Fsher equaton and the Benjamn-Bona-Mahony equaton by usng the drect algebrac method whch s ntroduced by Zhang [24]. 3. Applcatons Example 1. Consder Fsher equaton u t + u xx u + u 3 = 0. (4) For ths example we can use transformaton wth (1) and then (4) becomes kcu k 2 u u + u 3 = 0. (5) Balancng u 3 wth u gves m = 1. Therefore we may choose u = a 0 + a 1 F. (6) Substtutng (6) nto (5) yelds a set of algebrac equatons for a 0 a 1 a 2 k c and. These systems are found to be a 0 + a 3 0 a 1 ck = 0 a 1 + 3a 2 0a 1 2a 1 k 2 = 0 3a 0 a 2 1 a 1 ck = 0 a 3 1 2a 1 k 2 = 0. From the solutons of the system we have the followng two cases: Case 1. a 0 = 1 2 a 1 = 2 c = 3 k = (7) Case 2. a 0 = 1 2 a 1 = 2 c = 3 k = (8) By means of Mathematca substtutng (7) and (8) nto (6) we have obtaned the followng exact complex travelng wave solutons of (4). follows: Famly 1. u 1 = 1 2 These solutons are as 2 ( ( tanh 1 ) ( 2 2 x + 3 ))) 2

4 90 Kılıç and Baş where < 0 u 2 = 1 2 where < 0 where > 0 where > 0. Famly 2. u 5 = where < 0 where < 0 where > 0 where > 0. u 6 = ( ( coth 1 ) ( 2 2 x + 3 ))) 2 u 3 = 1 2 u 4 = 1 2 ( 2 tan 1 ( 2 x + 3 ))) 2 ( cot 1 ( 2 x + 3 ))) 2 ( ( tanh 1 ) ( 2 2 x 3 ))) 2 ( ( 2 coth 1 ) ( 2 2 x 3 ))) 2 u 7 = u 8 = ( 2 tan 1 ( 2 x 3 ))) 2 ( cot 1 ( 2 x 3 ))) Example 2. Consder the Benjamn-Bona-Mahony equaton u t + u x + uu x u xxt = 0. (9) For ths example f we use transformaton wth (1) then (9) becomes kcu + ku + kuu + 3 k 3 cu = 0 or equvalently u (1 c) + uu k 2 cu = 0. (10)

5 CUJSE 7 (2010) No Balancng uu wth u gves m = 2. Therefore we may choose u = a 0 + a 1 F + a 2 F 2. (11) Substtutng (11) nto (10) yelds a set of algebrac equatons for a 0 a 1 a 2 k c and. These systems are found to be a 1 + a 0 a 1 a 1 c 2a 1 ck 2 = 0 a a 2 + 2a 0 a 2 2a 2 c 16a 2 ck 2 = 0 a 1 + a 0 a 1 a 1 c + 3a 1 a 2 8a 1 ck 2 = 0 a a 2 + 2a 0 a 2 2a 2 c + 2a 40a 2 ck 2 = 0 3a 1 a 2 6a 1 ck 2 = 0 2a 24a 2 ck 2 = 0. From the solutons of the system we have the followng two cases: Case 1. a 0 = 1 + c + 8ck 2 a 1 = 0 a 2 = 12ck 2 ck 0 0. (12) Case 2. a 0 = 1 + c a 1 = 0 a 2 = 12ck 2 = 0. (13) By means of Mathematca substtutng (12) (13) nto (11) we have obtaned the followng exact complex travelng wave solutons of (9). These solutons are as follows: Famly 1. u 1 = 1 + c + 8ck ck ( 2 tanh ( k(x ct) )) 2 where < 0 where < 0 u 2 = 1 + c + 8ck ck 2 ( coth ( k(x ct) )) 2 where > 0 u 3 = 1 + c + 8ck ck 2 ( tan ( k(x ct) )) 2

6 92 Kılıç and Baş u 4 = 1 + c + 8ck ck ( 2 cot ( k(x ct) )) 2 where > 0. Famly 2. 1 u 5 = k(x ct) where = Concluson In ths paper we mplement a drect algebrac method [24] wth symbolc computaton to construct new exact complex solutons of the Fsher equaton and the Benjamn-Bona-Mahony equaton. The method can be used for many other nonlnear equatons. In addton ths method s computerzable whch allows us to perform complcated and tedous algebrac calculatons on a computer. References [1] L. Debtnath Nonlnear Partal Dfferental Equatons for Scentst and Engneers Brkhauser Boston MA [2] A. M. Wazwaz Partal Dfferental Equatons: Methods and Applcatons Balkema Rotterdam [3] W. Hereman P. P. Banerjee A. Korpel G. Assanto A. Van Immerzeele and A. Meerpoel Exact soltary wave solutons of nonlnear evoluton and wave equatons usng a drect algebrac method J. Phys. A 19 (1986) [4] A. H. Khater M. A. Helal and O. H. El-Kalaawy Bäcklund transformatons: exact solutons for the KdV and the Calogero-Degaspers-Fokas mkdv equatons Math. Meth. Appl. Sc. 21 (1998) [5] A. M. Wazwaz A study of nonlnear dspersve equatons wth soltary-wave solutons havng compact support Math. Comput. Smulaton 56 (2001) [6] S. A. Elwakl S. K. El-Labany M. A. Zahran and R. Sabry Modfed extended tanh-functon method for solvng nonlnear partal dfferental equatons Phys. Lett. A 299 (2002) [7] Y. Le Z. Fajang and W. Yngha The homogeneous balance method Lax par Hrota transformaton and a general ffth-order KdV equaton Chaos Soltons Fractals 13 (2002) [8] J. F. Zhang New exact soltary wave solutons of the KS equaton Int. J. Theor. Phys. 38 (1999) [9] M. L. Wang Exact solutons for a compound KdV-Burgers equaton Phys. Lett. A 213 (1996)

7 CUJSE 7 (2010) No [10] M. L. Wang Y. B. Zhou and Z. B. L Applcaton of a homogeneous balance method to exact solutons of nonlnear equatons n mathematcal physcs Phys. Lett. A 216 (1996) [11] M. L. Malflet Soltary wave solutons of nonlnear wave equatons Am. J. Phys. 60 (1992) [12] E. J. Parkes and B. R. Duffy An automated tanh-functon method for fndng soltary wave solutons to nonlnear evoluton equatons Comput. Phys. Commun. 98 (1996) [13] B. R. Duffy and E. J. Parkes Travellng soltary wave solutons to a seventh-order generalzed KdV equaton Phys. Lett. A 214 (1996) [14] E. J. Parkes and B. R. Duffy Travellng soltary wave solutons to a compound KdV-Burgers equaton Phys. Lett. A 229 (1997) [15] N. Bldk and H. Bayramoğlu The soluton of two dmensonal nonlnear dfferental equaton by the Adoman decomposton method Appl. Math. and Comput. 163 (2005) [16] T. Özş and A. Yıldırım Travelng wave soluton of Korteweg-de Vres equaton usng He s homotopy perturbaton method Int. J. Nonlnear Sc. and Numer. Smul. 8 (2007) [17] E. G. Fan Extended tanh-functon method and ts applcatons to nonlnear equatons Phys. Lett. A 277 (2000) [18] H. Chen and H. Zhang New multple solton solutons to the general Burgers-Fsher equaton and the Kuramoto-Svashnsky equaton Chaos Soltons Fractals 19 (2004) [19] H. Chen and H. Zhang New multple solton-lke solutons to the generalzed (2 + 1)- dmensonal KP equaton Appl. Math. Comput. 157 (2004) [20] Z. Y. Yan and H. Q. Zhang New explct soltary wave solutons and perodc wave solutons for Whtham-Broer-Kaup equaton n shallow water Phys. Lett. A 285 (2001) [21] E. G. Fan Auto-Bäcklund transformaton and smlarty reductons for general varable coeffcent KdV equatons Phys. Lett. A 294 (2002) [22] M. L. Wang and Y. M. Wang A new Bäcklund transformaton and mult-solton solutons to the KdV equaton wth general varable coeffcents Phys. Lett. A 287 (2001) [23] E. G. Fan and H. Q. Zhang New exact solutons to a system of coupled KdV equatons Phys. Lett. A 245 (1998) [24] H. Zhang A drect algebrac method appled to obtan complex solutons of some nonlnear partal dfferental equatons Chaos Soltons Fractals 39 (2009) [25] S. T. Mohyud-Dn A. Yıldırım and G. Demrl Travelng wave solutons of Whtham-Broer- Kraup equatons by homotopy perturbaton method Journal of Kng Saud Unversty-Scence. 22 (2010) [26] B. Ralar and A. Yıldırım The Applcaton of homotopy perturbaton method for MHD Flows of UCM Fluds above Porous Stretchng Sheets Comput. Math. Appl. 59 (2010) [27] A. Yıldırım S. T. Mohyud-Dn and D. H. Zhang Analytcal solutons to the pulsed Klen- Gordon equaton usng Modfed Varatonal Iteraton (MVIM) and Boubaker Polynomals Expanson Scheme (BPES) Comput. Math. Appl. 59 (2010)